Finite Group Databasebeta

← C3×#18317 #10636924

Subgroups — C3×#18317 #10636924

Order 768.

1402 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#534C211yesmaximal, characteristic
2C3×#3193C212yesmaximal, complemented
3C3×#2561C211yesmaximal, characteristic, complemented
4C3×#1607C214yesmaximal, complemented
5C3×#1601C214yesmaximal, complemented
6C3×#1507C212yesmaximal, complemented
7C3×#1462C211yesmaximal, characteristic, complemented
8#18317C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4—21no—
10C3×#534—24no—
11C3×#534—22no—
12C3×#534C2214yes—
13C2×C3×#123—22no—
14C2×C3×C22⋊Q8C2211yescharacteristic, complemented
15C2×C3×#121—21no—
16C2×C3×#121C2212yes—
17C2×C3×#120—22no—
18C22×C3×C22⋊C4C2211yescharacteristic
19C3×#484—22no—
20C3×#480C2214yescomplemented
21C3×#468—24no—
22C3×#450—22no—
23C3×#448C2214yescomplemented
24C3×#407C2212yescomplemented
25C3×#397C2212yescomplemented
26C3×#393C2212yescomplemented
27C3×#388—22no—
28C3×#381—21no—
29C3×#379—21no—
30C3×C4×C22⋊C4C2211yescharacteristic, complemented
31C3×#327C2218yescomplemented
32C3×#323C2214yescomplemented
33C2×#534C2×C311yescharacteristic
34#3193C2×C312yescomplemented
35#2561C2×C311yescharacteristic, complemented
36#1607C2×C314yescomplemented
37#1601C2×C314yescomplemented
38#1507C2×C312yescomplemented
39#1462C2×C311yescharacteristic, complemented
40C25×C3—21no—
41C25×C3D411yescharacteristic
42C22×C3×Q8C2311yescharacteristic, complemented
43C22×C3×D4—22no—
44C22×C3×D4—24no—
45C22×C3×D4—22no—
46C22×C3×D4—24no—
47C22×C3×D4—21no—
48C22×C3×D4—48no—
49C22×C3×D4D412yes—
50C23×C3×C4—21no—
51C23×C3×C4D411yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C2×C3×SD16—24no—
54C2×C3×M16—22no—
55C3×#123—28no—
56C3×#123—28no—
57C3×C22⋊Q8—22no—
58C3×C22⋊Q8—22no—
59C3×#121—24no—
60C3×#121—24no—
61C3×#121—22no—
62C3×#121—22no—
63C3×#121D414yes—
64C3×#121C2314yes—
65C3×#120—28no—
66C3×#120—28no—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4D412yes—
70C2×C3×C4⋊C4C2311yescharacteristic, complemented
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4—22no—
79C2×C3×C22⋊C4D412yes—
80C2×C3×C22⋊C4C2312yes—
81C2×C3×C42—21no—
82C3×#107—24no—
83C3×#103—24no—
84C3×#102—24no—
85C3×C22⋊C8—22no—
86C3×C22⋊C8C2314yescomplemented
87C3×#95—22no—
88C3×#95—22no—
89C3×#95C2312yescomplemented
90C23×D4—21no—
91#534—24no—
92#534—22no—
93#534C22×C314yes—
94C2×#123—22no—
95C2×C22⋊Q8C22×C311yescharacteristic, complemented
96C2×#121—21no—
97C2×#121C22×C312yes—
98C2×#120—22no—
99C22×C22⋊C4C22×C311yescharacteristic
100#484—22no—
101#480C22×C314yescomplemented
102#468—24no—
103#450—22no—
104#448C22×C314yescomplemented
105#407C22×C312yescomplemented
106#397C22×C312yescomplemented
107#393C22×C312yescomplemented
108#388—22no—
109#381—21no—
110#379—21no—
111C4×C22⋊C4C22×C311yescharacteristic, complemented
112#327C22×C318yescomplemented
113#323C22×C314yescomplemented
114C24×C3—21no—
115C24×C3—21no—
116C24×C3—24no—
117C24×C3—24no—
118C24×C3—22no—
119C24×C3—44no—
120C24×C3—42no—
121C24×C3—81no—
122C24×C3—82no—
123C24×C3—82no—
124C24×C3C2×D411yescharacteristic
125C24×C3C2×D412yes—
126C2×C3×Q8—22no—
127C2×C3×Q8—81no—
128C2×C3×D4—28no—
129C2×C3×D4—24no—
130C2×C3×D4—24no—
131C2×C3×D4—24no—
132C2×C3×D4—22no—
133C2×C3×D4—48no—
134C2×C3×D4—416no—
135C2×C3×D4—48no—
136C2×C3×D4—48no—
137C2×C3×D4—48no—
138C2×C3×D4—82no—
139C2×C3×D4—82no—
140C2×C3×D4C2×D414yes—
141C22×C3×C4—24no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—44no—
149C22×C3×C4—42no—
150C22×C3×C4—81no—
151C22×C3×C4C2×D412yes—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2411yescharacteristic
156C3×SD16—84no—
157C3×M16—82no—
158C2×C3×C8—24no—
159C3×C4⋊C4—24no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—42no—
163C3×C4⋊C4C2×D414yes—
164C3×C22⋊C4—28no—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—24no—
168C3×C22⋊C4—24no—
169C3×C22⋊C4—42no—
170C3×C22⋊C4—41no—
171C3×C22⋊C4—41no—
172C3×C22⋊C4—44no—
173C3×C22⋊C4—82no—
174C3×C42—81no—
175C25—21no—
176C25C3×D411yescharacteristic
177C22×Q8C23×C311yescharacteristic, complemented
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—22no—
181C22×D4—24no—
182C22×D4—21no—
183C22×D4—48no—
184C22×D4C3×D412yes—
185C23×C4—21no—
186C23×C4C3×D411yescharacteristic
187C23×C4C23×C311yescharacteristic
188C2×SD16—24no—
189C2×M16—22no—
190#123—28no—
191#123—28no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194#121—24no—
195#121—24no—
196#121—22no—
197#121—22no—
198#121C3×D414yes—
199#121C23×C314yes—
200#120—28no—
201#120—28no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C3×D412yes—
205C2×C4⋊C4C23×C311yescharacteristic, complemented
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4C3×D412yes—
215C2×C22⋊C4C23×C312yes—
216C2×C42—21no—
217#107—24no—
218#103—24no—
219#102—24no—
220C22⋊C8—22no—
221C22⋊C8C23×C314yescomplemented
222#95—22no—
223#95—22no—
224#95C23×C312yescomplemented
225C23×C3—24no—
226C23×C3—24no—
227C23×C3—22no—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—24no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3—42no—
236C23×C3—44no—
237C23×C3—48no—
238C23×C3—44no—
239C23×C3—42no—
240C23×C3—44no—
241C23×C3—81no—
242C23×C3—81no—
243C23×C3—88no—
244C23×C3—84no—
245C23×C3—84no—
246C23×C3—816no—
247C23×C3#12011yescharacteristic
248C23×C3#12011yescharacteristic
249C23×C3AGL(1,ℤ/8)14yes—
250C23×C3C22×D411yescharacteristic
251C3×Q8—44no—
252C3×D4—44no—
253C3×D4—416no—
254C3×D4—48no—
255C3×D4—48no—
256C3×D4—82no—
257C3×D4—84no—
258C2×C3×C4—22no—
259C2×C3×C4—24no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—44no—
265C2×C3×C4—48no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—81no—
276C2×C3×C4#12012yes—
277C2×C3×C4C22×D412yes—
278C3×C8—44no—
279C24—21no—
280C24—21no—
281C24—24no—
282C24—24no—
283C24—22no—
284C24—44no—
285C24—42no—
286C24—81no—
287C24—82no—
288C24—82no—
289C24C2×C3×D411yescharacteristic
290C24C2×C3×D412yes—
291C2×Q8—22no—
292C2×Q8—81no—
293C2×D4—28no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—22no—
298C2×D4—48no—
299C2×D4—416no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—48no—
303C2×D4—82no—
304C2×D4—82no—
305C2×D4C2×C3×D414yes—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—44no—
314C22×C4—42no—
315C22×C4—81no—
316C22×C4C2×C3×D412yes—
317C22×C4C2×C3×D411yescharacteristic
318C22×C4C2×C3×D412yes—
319C22×C4C2×C3×D412yes—
320C22×C4C24×C311yescharacteristic
321SD16—84no—
322M16—82no—
323C2×C8—24no—
324C4⋊C4—24no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—42no—
328C4⋊C4C2×C3×D414yes—
329C22⋊C4—28no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—44no—
338C22⋊C4—82no—
339C42—81no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—24no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—48no—
351C22×C3—42no—
352C22×C3—42no—
353C22×C3—44no—
354C22×C3—44no—
355C22×C3—44no—
356C22×C3—82no—
357C22×C3—88no—
358C22×C3—84no—
359C22×C3—816no—
360C22×C3#45512yes—
361C22×C3C2×#12011yescharacteristic
362C22×C3C2×AGL(1,ℤ/8)12yes—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—44no—
366C3×C4—42no—
367C3×C4—82no—
368C3×C4—81no—
369C3×C4—81no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—24no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—44no—
382C23—48no—
383C23—44no—
384C23—42no—
385C23—44no—
386C23—81no—
387C23—81no—
388C23—88no—
389C23—84no—
390C23—84no—
391C23—816no—
392C23C3×#12011yescharacteristic
393C23C3×#12011yescharacteristic
394C23C3×AGL(1,ℤ/8)14yes—
395C23C22×C3×D411yescharacteristic
396Q8—44no—
397D4—44no—
398D4—416no—
399D4—48no—
400D4—48no—
401D4—82no—
402D4—84no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—81no—
421C2×C4C3×#12012yes—
422C2×C4C22×C3×D412yes—
423C8—44no—
424C2×C3—22no—
425C2×C3—22no—
426C2×C3—41no—
427C2×C3—44no—
428C2×C3—84no—
429C2×C3#300311yescharacteristic
430C2×C3C2×#45511yescharacteristic
431C2×C3#302611yescharacteristic
432C22—21no—
433C22—21no—
434C22—21no—
435C22—21no—
436C22—24no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—44no—
441C22—44no—
442C22—48no—
443C22—42no—
444C22—42no—
445C22—44no—
446C22—44no—
447C22—44no—
448C22—82no—
449C22—88no—
450C22—84no—
451C22—816no—
452C22C3×#45512yes—
453C22C2×C3×#12011yescharacteristic
454C22C2×C3×AGL(1,ℤ/8)12yes—
455C4—22no—
456C4—22no—
457C4—44no—
458C4—42no—
459C4—82no—
460C4—81no—
461C4—81no—
462C3#1831711yescharacteristic, direct product factor, Sylow
463C2—22no—
464C2—22no—
465C2—41no—
466C2—44no—
467C2—84no—
468C2C3×#300311yescharacteristic
469C2C2×C3×#45511yescharacteristic
470C2C3×#302611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#534C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C3×#3193C2semidirect
3. C3×#2561C2semidirect
4. C3×#1607C2semidirect
5. C3×#1601C2semidirect
6. C3×#1507C2semidirect
7. C3×#1462C2semidirect
8. #18317C3direct (N × H)
14. C2×C3×C22⋊Q8C22semidirect
20. C3×#480C22semidirect
23. C3×#448C22semidirect
24. C3×#407C22semidirect
25. C3×#397C22semidirect
26. C3×#393C22semidirect
30. C3×C4×C22⋊C4C22semidirect
31. C3×#327C22semidirect
32. C3×#323C22semidirect
34. #3193C2×C3semidirect
35. #2561C2×C3semidirect
36. #1607C2×C3semidirect
37. #1601C2×C3semidirect
38. #1507C2×C3semidirect
39. #1462C2×C3semidirect
42. C22×C3×Q8C23semidirect
70. C2×C3×C4⋊C4C23semidirect
86. C3×C22⋊C8C23semidirect
89. C3×#95C23semidirect
95. C2×C22⋊Q8C22×C3semidirect
101. #480C22×C3semidirect
104. #448C22×C3semidirect
105. #407C22×C3semidirect
106. #397C22×C3semidirect
107. #393C22×C3semidirect
111. C4×C22⋊C4C22×C3semidirect
112. #327C22×C3semidirect
113. #323C22×C3semidirect
177. C22×Q8C23×C3semidirect
205. C2×C4⋊C4C23×C3semidirect
221. C22⋊C8C23×C3semidirect
224. #95C23×C3semidirect
462. C3#18317direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C3×#5342
C3×#31932
C3×#25612
C3×#16072
C3×#16012
C3×#15072
C3×#14622
#183173